A067276 Determinant of n X n matrix containing the first n^2 primes in increasing order.

2, -1, -78, 880, -4656, -14304, -423936, 8342720, 711956736, -615707136, 21057138688, -4663930678272, 211912980656128, -9178450735677440, 40005919124799488, 83013253447139328, -8525111273818357760, -800258888289188708352, -15170733077495639179264

Offset

1,1

Comments

The first column contains the first n primes in increasing order, the second column contains the next n primes in increasing order, etc. Equivalently, first row contains first n primes in increasing order, second row contains next n primes in increasing order, etc. Sequences of determinants of matrices specifically containing primes include A024356 (Hankel matrix), A067549 (first n primes on diagonal, other elements 1), A066933 (cyclic permutations of first n primes in each row) and A067551 (first n primes on diagonal, other elements 0).

Links

Robert Israel, Table of n, a(n) for n = 1..459

Example

a(3) = -78 because det[[2,7,17],[3,11,19],[5,13,23]] = -78 (= det[[2,3,5],[7,11,13],[17,19,23]], the determinant of the transpose.).

Maple

seq(LinearAlgebra:-Determinant(Matrix(n, n, (i, j) -> ithprime(n*(i-1)+j))), n=1..20); # Robert Israel, Jul 12 2017

Mathematica

Table[ Det[ Partition[ Array[Prime, n^2], n]], {n, 19}] (* Robert G. Wilson v, May 26 2006 *)

Prog

(PARI) for(n=1, 20, k=0; m=matrix(n, n, x, y, prime(k=k+1)); print1(matdet(m), ", ")) /* The matrix initialization command above fills columns first: Variables (such as) x and y take on values 1 through n for rows and columns, respectively, with x changing more rapidly and they must be specified even though the 5th argument is not an explicit function of them here. */
(Magma) [ Determinant( Matrix(n, n, [ NthPrime(k): k in [1..n^2] ]) ): n in [1..19] ]; // Klaus Brockhaus, May 12 2010
(Python)
from sympy.matrices import Matrix
from sympy import sieve
def a(n):
    sieve.extend_to_no(n**2)
    return Matrix(n, n, sieve[1:n**2+1]).det()
print([a(n) for n in range(1, 20)]) # Indranil Ghosh, Jul 31 2017

Crossrefs

Cf. A024356, A067549, A066933, A067551, A114533.

Cf. A119894, A118770, A118772, A118779.

Sequence in context: A216485 A096681 A247793 * A118580 A118558 A095837

Adjacent sequences: A067273 A067274 A067275 * A067277 A067278 A067279

Keyword

easy,sign

Author

Rick L. Shepherd, Feb 21 2002

Status

approved